notes

New Economic School, 2005/6FINANCIAL RISK MANAGEMENT

Lecture notes

Plan of the course Classification of risks and strategic risk management Derivatives and financial engineering Market risk Liquidity risk Credit risk Operational risk

Lecture 1. Introduction

Plan Definition of risk Main types of risks Examples of financial failures Specifics of financial risk management Empirical evidence on RM practices

What is risk? Chinese hyeroglif “risk”

o Danger or opportunity o This is the essence of financial risk-management!

Uncertainty vs risko Subjective / objective probabilitieso Speculative / pure

How to measure risk? Probability / magnitude / exposure

o Systematic vs residual risk Maximal vs average losses Absolute vs relative risk

How to classify risks? Nature / political / transportation / … Commercial

o Property / production / trade / …o Financial

Investment: lost opportunity (e.g. due to no hedging), direct losses, lower return Purchasing power of money: inflation, currency, liquidity

Main types of financial risks Market risk

o Interest rate / currency / equity / commodity Credit risk

o Sovereign / corporate / personal Liquidity risk

o Market / funding Operational risk

Financial Risk Management, NES 2006/7

o System & control / management failure / human error Event risk

Examples of financial failures

Lessons for risk management Integrated approach to different types of risks Portfolio view Accounting for derivatives Market microstructure Role of regulators and self-regulating organizations

Methods for dealing with uncertainty by Knight Consolidation Specialization Control of the future Increased power of prediction

Financial risk management Avoid?

o But you cannot earn money without taking on risks Reserves: esp. banks Diversification

o But: only nonsystematic risk Hedging

o Usually, using derivatives Insurance: for exogenous low-probability events

o Otherwise bad incentives Evaluation based on risk-adjusted performance Strategic RM: enterprise-wide policy towards risks

o Identification / Measurement / Management / Monitoring

Should the companies hedge? NO The MM irrelevance argument

o The firm’s value is determined by its asset side

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The CAPM argumento Why hedge unsystematic (e.g., FX) risk?o Any decrease in % will be accompanied by decrease in E[CF]!

Transactions with derivatives have negative expected value for a companyo After fixed costs

Should the companies hedge? YES Both MM and CAPM require perfect markets

o Bankruptcy costs are important The CAPM requires diversification

o Real assets are not very liquid and divisible Shareholder wealth maximization

o Market frictions: financial distress costs / taxes / external financing costs Managerial incentives

o Improving executive compensation and performance evaluation Improving decision making

Empirical evidence on RM practices Financial firms:

o The size of derivative positions is much greater than assets (often more than 10 times) Non-financial firms:

o Main goal: stabilize CFso Firms with high probability of distress do not engage in more RMo Firms with enhanced inv opportunities and lower liquidity are more likely to use derivatives

Methods for dealing with uncertainty by Knight Consolidation Specialization Control of the future Increased power of prediction

Financial risk management Avoid?

o But you cannot earn money without taking on risks Reserves: esp. banks Diversification

o But: only nonsystematic risk Hedging

o Usually, using derivatives Insurance: for exogenous low-probability events

o Otherwise bad incentives Evaluation based on risk-adjusted performance Strategic risk management: enterprise-wide policy towards risks

o Identification / Measurement / Management / Monitoring

Current trends in risk management Deregulation of financial markets Increasing banking supervision and regulation Technological advances Results: risk aggregation, increasing systemic and operating risks

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Lecture 2. Financial engineering

Plan Specifics of risks of different instruments

o Investment strategies / pricing / systematic risks Stocks / bonds / derivatives

o Forwards / futures / options / swaps

General approach to financial risk modeling Use of returns

o Stationary (in contrast to prices) Risk mapping: projecting our positions to (a small set of) risk factors

o We might not have enough observations for some positions E.g., new market or instrument

o Too large dimensionality of the covariance matrix For n assets: n variances and n(n-1)/2 correlations

o Excessive computations during simulations

Specifics of risks for different assets Discounted cash flow approach: P0 = Σt CFt/(1+r)t

Stocks: P0 = (P1+Div1)/(1+r) = Σt=1:∞ Divt/(1+r)t

o Interest rates / Exchange rates o Prices on goods and resourceso Corporate governance / Political risk

Bonds: P0 = Σt=1:T C/(1+rt)t + F/(1+rT)T

o Interest rates for different maturitieso Default risk

Derivativeso Price of the underlying asset

Shape of the payoff function Volatility

o Interest rates

Index models: Ri,t = αi + ΣkβkiIk

t + εi,t, where E(εi,t)=0, cov(Ik, εi)=0, and E(εiεj)=0 for i≠j Risk management: ΔRi ≈ Σkβk

iΔIk

Separation of total risk on systematic and idiosyncratic: var(Ri) = βi2σ2

M + σ2(ε)i

o Systematic risk depends on factor exposures (betas): βi2σ2

M

o Idiosyncratic risk can be reduced by diversification Covariance matrix: cov(Ri, Rj) = βiβjσ2

M o Correlations computed directly from the historical data are bad predictors

StocksSingle-index model with market factor: Ri,t = αi + βiRM

t + εi,t

(Market model, if we don’t make an assumption E(εiεj)=0 for i≠j)where β=cov(Ri, RM)/var(RM): (market) beta, sensitivity to the market risk

Multi-index models: Industry indices Macroeconomic factors

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Salomon Brothers: -Economic growth-Default spread-LR interest rates-SR interest rates-Inflation shock-US dollar


o Oil price, inflation, exchange rates, interest rates, GDP/ consumption growth rates Investment styles

o Small-cap / large-capo Value / growth (low/high P/E)o Momentum / reversal

Statistical factorso Principal components

Investment strategies Speculative: choosing higher beta

o Increases expected return and risko Used by more aggressive mutual funds

Hedging (systematic risk): β = 0o Market-neutral strategy: return does not depend on the market movemento Often used by hedge funds

Arbitrage: riskless profit (“free lunch”)o Buy undervalued asset and sell overvalued asset with the same risk characteristicso Pure arbitrage is very rare: there always some risks

Bonds

Single-index model with interest rate: Ri,t = ai + Di Δyt + ei,t

where yt: interest rate in period t, D: duration, exposure to interest risk

Macauley duration: D = -[∂P/P]/[∂y/y] = -Σt=1:T t Ct / (P yt)For the bond with the price: P0 = Σt=1:T Ct / yt Wtd-average maturity of bond payments, D ≤ T Elasticity of the bond’s price to its YTM (yield to maturity) For small changes in %: ΔP/P ≈ -D Δy/y = -D* Δy

o D* = D/y: modified durationConvexity: C = -Σt=1:T t(t+1)Ct / (P yt

t) For small changes in %: ΔP/P ≈ -D Δy/y + ½ C (Δy/y)2

Asset-liability management: used by pension funds, insurance companies Gap analysis: gapt = At-Lt

o Positive gap implies higher interest income in case of rising % Perfect hedging: zero gaps (cash flow matching)

o Can be unachievable or too expensive Immunization: D(assets) = D(liabilities)

o Active strategy, since both duration and the term structure of interest rates evolve over timeo Need precise measure of duration (and convexity)o Does not protect from large changes in %

Derivatives

Derivatives: Unbundled contingent claims

o Forwards / Futures / Swaps / Options Embedded options:

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o Convertible / redeemable bonds Role of derivatives: efficient risk sharing

o Speculation: give high leverageo Hedging: reduce undesirable risks

Notional size: around $140 trln o Twice as large as equity and bond markets combined

The total market value (based on positive side): less than $3 trln

Forward / futures Obligation to buy or sell the underlying asset in period T at fixed settlement price K Zero value at the moment of signing the contract (t=0) Payoff at T, long position: ST-F

Forward Specific terms Spot settlement Low liquidity

o Must be offset by the counter deal Credit risk

Futures Standardized exchange-traded contract

o Amount, quality, delivery date, place, and conditions of the settlement Credit risk taken by the exchange

o The exchange clearing-house is a counter-partyo Collateral: the initial / maintenance margino Marking to market daily

Long position: receive A(Ft-Ft-1) into account High liquidity, popular among speculators

o Can be offset by taking an opposite positiono Usually, cash settlement

No-arbitrage forward price F (assuming perfect markets): For assets with known dividend yield q: F = Se(r-q)T

o Value of the long position: (F-K)e-rT = Se-qT -K-rT

Systematic risks Delta (first derivative wrt the price of the underlying): δ=e-qT

Gamma (second derivative wrt the price of the underlying): zero!

Specifics of futures If r=const, futures price = forward price If r is stochastic and corr(r, S)>0, futures price > forward price

o The margin proceeds will be re-invested at higher rate Liquidity risk due to margin requirements Basis risk: the basis = spot price – futures price

o Ideal hedge: the basis=0 at the delivery dateo Usually, the basis > 0 at the settlement date

Maturity / quality / location risks

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Example: Metallgesellschaft Sold a huge volume of 5-10 year oil forwards in 1990-93, hedging with short-term futures When the oil price fell, the margin requirements exceeded $1 bln. The Board of Directors decided

to fix the futures’ losses and close forward positions. The final losses were around $1.3 bln. Lessons:

o The rollover basis risk was ignored by those managers who designed the strategyo The senior management did not understand this strategy and therefore made clearly inefficient

decision to close long forward positions that were profitable after decline in oil prices.

Investment strategies Speculative

o Naked: buying or selling futureso Spread: calendar / cross

Hedging o E.g., short hedge: we need to sell the underlying asset, hedge with short futures

Hedge ratio: hedged position / total positiono Hedging stock exposure with stock index futures: βSo Hedging interest rate risk with duration: immunization

Options: European call (put): right to buy (sell) the underlying asset at the exercise date T at the

strike/exercise price K American call (put): can be exercised at any time before T Right, no obligation (for the buyer) => asymmetric payoff function

o Call: cT = max(ST-K, 0)o Put: pT = max(K-ST, 0)

Synthetic forward: long call, short put European call-put parity: c0 + Ke-rT = p0 + S0

o Covered put = call + cash

Speculative strategies Naked / covered option Spread: options of one type

o Bear / bull: long and short call (put)o Butterfly: long with K1 and K3, two short with K2= ½ (K1+K3)o Calendar: short with T and long with T+t with the same strike

Combination: options of different typeo Straddle: call and puto Strip: call and two putso Strap: two calls and put o Strangle: with different strikes

Black-Scholes model Call: ct = Ste-qT N(d1) – Xe-rT N(d2) Put: p = Xe-rT N(-d2) – Se-qT N(-d1)

o d1 = [ln(S/X) + T(r-q+σ2/2)] / [σ√T], d2 = d1 – σ√To q is cont. dividend yieldo N(.) is a std normal distribution function

Given price, σ is implied volatilityo Good forecast of future volatility of the underlying

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Systematic risks: the greeks Delta (wrt price of the underlying asset)

o Call: δ =e-qT N(d1)o Put: δ=-e-qT N(-d1)

Rho (wrt risk-free rate)o Call: ρ=XTe-rT N(d2) o Put: ρ=-XTe-rT N(-d2)

Vega (wrt volatility) Theta (wrt time)

Hedging strategies Delta-neutral Gamma-neutral Delta-rho-neutral

Swaps Interest rate swap: exchange of fixed-rate and floating-rate interest payments for a fixed par value

o Sensitive to interest rate risko Pricing swap: via decomposition of PV(fixed coupons) and PV(forward rate coupons)

The market price of the floating-rate bond equals par after each coupon payment! Currency swap: exchange of interest payments in different currencies

o Sensitive to interest rate and currency risks

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Lecture 3. Measuring volatility

Historical volatility: MA Moving Average with equal weights

EWMA (used by RiskMetrics)σ2

t = λσ2t-1 + (1-λ)r2

t-1 = (1-λ) Σk>0 λt-1r2t-k

Exponentially Weighted Moving Average quickly absorbs shocks λ is chosen to minimize Root of Mean Squared Error

RMSE = √ (1/T)∑t=1:T (σ2t-r2

t)2

λ = 0.94 for developed markets

GARCH(1,1)σ2

t = a + b σ2t-1 + cε2

t-1

Parameter restrictions: a>0, b+c<1o Guarantee that variation is non-negative and that unconditional expectation exists:

E[σ2] = a/(1-b-c) More general model than EWMA, can be modified

o More lagso Leverage effect: stronger reaction to negative shocks

More parameters leads to larger estimation erroro Used less frequently in RM than EWMA

Implied Based on options’ market prices and (Black-Sholes) model

o Forward-looking!

Realized Based on intraday data

o E.g., prices over hourly intervalso Only for liquid assets

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How are size and direction of the position related to the liquidity risk?


Lectures 4-7. Market risk: VaR and beyond

Identification of market risk Primary: directional risks from taking a net long/short position in a given asset class

o Interest rate / currency / equity / commodity Secondary: other

o Volatility / spread / dividendo Many trading books are managed with the objective of reducing primary risks…

at the expense of an increase in secondary risks General (systematic) vs specific risks

o The latter may be important because of lack of diversification or implementation of a specific investment strategy

Non-linear (option-like) instrumentso Need to model full probability distribution of the underlying markets factor(s)o Sensitive to long-term market volatilities and correlations

Assessment of market risk: selection of factors and choice of models Statistical models: factor return distributions

o Describing uncertainty about the future values of market factorso E.g., geometric Brownian motion, GARCH

Pricing models: factor exposureso Relating the prices and sensitivities of instruments to underlying market factorso E.g., Black-Scholes

Risk aggregation models: risk measureso Evaluating the risk of losses in the future portfolio’s valueo E.g., standard deviation, VaR

Traditional measures of market risk Position

o Size and direction Volatility

o Time aggregation: √T ruleo Cross aggregation: diversification effect

Exposureo Stocks: betao FI: duration, convexityo Derivatives: delta, gamma, vega, ...

Issues Aggregation and comparability of risks

o Can’t sum up deltas or vegaso Market vs credit vs operational risk

Measuring losses Controlling risk: position limits

o Until 1990s, mostly restrictions on size of net positions, including delta equivalent exposures

History of Market Risk Management In late 1970s and 1980s,

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How does VaR change with holding period / confidence interval?


o Major financial institutions started work on internal models to measure and aggregate risks across the institution.

o As firms became more complex, it was becoming more difficult but also more important to get a view of firm-wide risks

o Firms lacked the methodology to aggregate risks from sub-firm level Early 1990s

o Group of 30 report. Derivatives: Practices and Principles. Their work helped shape the emerging field of financial risk management

o Oct 1994 JP Morgan published RiskMetrics and made the data and methodology freely available on the internet. Riskmetrics was developed over the previous 8 months, based on their own internal model.

1995-2005o Value at Risk (VaR) becomes a standard financial market risk measurement tool worldwide

Value-at-Risk (VaR) Maximum loss due to market fluctuations over a certain time period with a given probability 1-α:

Prob(Loss<VaR)=1-α Key parameters:

o Confidence level: 99% (Basel) or 95% (RiskMetrics) The higher the confidence level, the lower the precision

o Holding period: 10d (Basel) or 1d (RiskMetrics) Time necessary to close or hedge the position Investment horizon

Estimation period Statistical model

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VaR measurement: general framework Marking-to-market position Portfolio sensitivity to risk factors

o Linear vs non-linear Distribution of the risk factors

o Normal vs other Parameter assumptions

o Confidence level and horizon Data

o Estimation period and frequency

Main methods of computing VaR Delta-normal

o Analytic, variance-covariance Historical simulation

o Bootstrap Monte Carlo

o Simulations Stress Testing and Scenario Analysis are complementary tools to VaR

o Focus on potential extreme market moves

Local estimation approaches

Standard delta-normal method Assuming that ptf returns are normally distributed: VaR = V(1-exp(k1-ασt+μ)) ≈ k1-αVσt

o Quantile: 1.65 (95%) и 2.33 (99%)o Daily data: assume that expected return = 0

Holding period up to 10 dayso T-day Var = daily Var * √To Assuming zero auto-correlation

Applications:o Single asseto Large, well-diversified ptf of many iid positions (e.g., consumer credits)o ‘Quick and dirty’ way to compute VaR of the business unit, based on historical P&L

Delta-normal method: risk mapping Decomposition of the ptf to multiple risk factors: V = ΣiWiFi

VaR = k1-αV√WTΣWo V is decomposed based on Taylor serieso W: vector of ptf weights or sensitivities of ptf return to factor returnso Σ: covariance matrix of risk factors

Can be decomposed to SD and correlations: covi,j = corri,j σi σj

Correlations are more stable in time, use larger estimation period SD is more time-varying, computed by EWMA or GARCH

Decomposition of the ptf to standardized positions Xi

VaR = k1-α√XTΣXo Standardized positions: sensitive only to the given factor, with same delta as the ptf

Example: for a US investor, ptf of dollar bonds on $2 mln has std positions for interest rate risk on $2 mln and for FX risk on $2 mln

o Two-stage approach by RiskMetrics: VaR = √ PVaRT Ω PVaR

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Will delta-normal method under- or overstate risk in presence of options?


Estimate risks of each std position: PVaR Aggregate using pre-estimated correlation matrix of std positions Ω

Dealing with deviations from normal distribution: Fat tails / skewness

o Adjusted quantiles based on Student’s t distribution or mixture of normal distributions Nonlinear relationships: dV = Δ dS + ½ Γ dS2 +Λ dσ + Θ dt + ...

o Options: deltas are unstable and asymmetrico Delta-gamma-vega approximation: VaR = |Δ| k1-ασS – ½ Γ (k1-ασS)2 + |Λ| |dσ|

Critique Quick computations Decomposes risks Strong assumptions about return distributions Cannot handle complicated derivatives The computations rise geometrically with # factors Can’t aggregate volatility over time with √T for large T

Full estimation approaches

Historical simulation method: nonparametric, full estimation approach Take historical time series of risk factors or assets

o Usually, at least one yearo Longer period increases the precision unless the process properties change over time

Simulate the change in value of a given ptf using historical factor realizationso Actual price functions (e.g., Black-Scholes)o Approximate ptf payoff function (based on ptf sensitivities)

Repeat simulations, plot the empirical distribution of P&L

Modifications: Different probabilities for historical observations

o EWMA: geometrically decreasing probabilities λt for lag t (0<λ<1) More distant events have smaller probability

o Higher weight for observations from the same month For seasonal commodities, such as natural gas

Hull-White, standardized returns: use Ri,tsi/σi,t in the simulationso σi,t: historical volatility of factor io si: current volatility forecast for factor i

Intra-day returns (e.g., hourly intervals)o Can analyze assets with short history (e.g., after IPO)

Critique Easy and simple No model risk

o No need to assume normal distribution, forecast volatility Correlations are embedded Path-dependent, assumes stationarity Requires long history

o Otherwise miss rare shocks Does not give structural knowledge

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Monte Carlo (simulation) method Model (multivariate) factor distributions

o Stocks: Geometric Brownian Motion / with jumpso Interest rates: Vasicek / CIR / multifactor models

Generate scenarios and compute the realized P&Lo Using factor innovations from the model

Plot the empirical distribution of P&L

Critique Most powerful and flexible

o (Cross-)factor dependencieso Most complicated instruments (path-dependent options)

Intellectual and technological skills required Lengthy computations Model risk

West (2004), Comparative summary of the methods

Different types of VaR VaR delta: partial derivative wrt factor or position VaR beta: % measuring the contribution of a given factor or position to the overall ptf risk Incremental VaR: change in VaR due to a change in the position

o Precise measurement requires re-estimation Marginal (component) VaR: Delta*Position

o Additive: ptf VaR is a sum of marginal VaRs Relative VaR: VaR of the ptf’s deviation from the benchmark

o Measures excessive risk

Backtesting VaR Verification of how precisely VaR is measured

o Compare % cases when the losses exceed VaR with the predicted frequency Historical approach: based on the actual recorded P&L

o More traditional, required by Baselo Helps to identify the model’s weaknesses, mistakes in the data, and intra-day tradingo Often, actual P&L produces lower than expected frequency of VaR violations due to day

trading that allows positions to be closed quickly when the markets become volatile Thus, reducing actual losses compared holding a static portfolio for 24 hours

Hypothetical approach: based on hypothetical P&L computed using current ptf weights (factor exposures) and historical data on assets (risk factors)o Concentrates on the current risk profileo Eliminates the impact of intra-day trading

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Should VaR be conservative?


o But: may give biased results if the same model is used both for estimating P&L and for VaR Small sample problem

o Need long history for high confidence level (99%) to ensure statistical accuracy of VaR Basel: 1 year of daily data

# exceptions Zone Scaling factor for reserves0-4 Green 35 Yellow 3.46 Yellow 3.57 Yellow 3.658 Yellow 3.759 Yellow 3.8510+ Red 4 (model withdrawn)

Berkowitz and O’Brien, JF 2002 Examine the practice of VaR measurement in 6 large US banks

o Compare VaR based on internal model with that based on reduced-form model for P&L Data, 01/1998 – 03/2000

o Consolidated end-of-day P&L, internal daily 99% VaRo The returns are normalized by SD to hide the identities of the banks

Т1: for 5 from 6 banks VaR is exceeded in 3 or less cases from 570 Т1, F1: when VaR is exceed, the losses are high, over 2 SD for 3 banks (prob. 0.1% for t5) Т2, F2: most cases of exceeding VaR during the 3 months of the Russian default in 8-10/1998

o Banks make conservative estimates of VaR: the exceptions are rare, but large and clustered in time

Т3: the correlation between banks’ daily P&L is quite low (0.2), the correlation between banks’ daily VaR is unstable

Т4: evaluating the accuracy of VaR estimateso Unconditional coverage test: reject H0 for one bank (but low power)o Independence (for i.i.d. distribution of violations): reject H0 for 2 bankso Conditional coverage test (sum of the previous two): reject H0 for 2 banks

Compare with the reduced-form models: ARMA(1,1) & GARCH(1,1), out-of-sample starting after 165 dayso Does not account for change in positions and risks, cannot be used for sensitivity or scenario

analysiso Embeds systematic mistakes in P&Lo Т5, F4: VaR is lower, the average # exceptions close to 1%, the magnitude of losses is lower,

similar test results Conclusions:

o Banks’ estimates of VaR are too conservative, do not adequately reflect risks in certain periods; are not better than those based on a simpler forecasting models (ARMA-GARCH)

o Makes sense to use both models Traditional: forward-looking, decomposes individual risks Times series: flexible and parsimonious, advantage in forecasting, provides check for the

main model

VaR implementation Measurement

o Local vs full estimationo Portfolio effects

Applications Verification: back testing Sensitivity: stress testing

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Why does Basel accord set the value of multiplier to 3?


Limitations

VaR applications Portfolio management

o Min VaR with given expected return Position limits

o Trading vs investmento Hierarchical structure

Capital adequacy requirementso Basel: reserves = 3*VaR99%

o The multiplier goes up if VaR is underestimated Risk-adjusted performance evaluation

o RAROC = Risk-Adjusted Return / Economic Capitalo Similar measures in corporate finance: EaR (Earnings at Risk), CFaR (Cash Flow at Risk)

Stress testing Analysis of ptf value under rare, but possible scenario

o Shows the magnitude of losses which exceed VaRo Helps to identify the weaknesses

Factor push: assume that one of the key model parameters changes a loto E.g., increase in volatility / correlations / exchange rate / oil priceo For complicated derivatives need to check how the ptf value changes under different values of

the parametero But: correlations are ignored

Historical scenarioso August 1998: Russian gvt default, ruble devaluation, credit spreads rising, developed

countries’ bonds rates falling, gradual liquidity crisiso Shock for a certain asset class: Black Monday for the US stock market in 1987, war in Iraq and

oil price, etc.o Shock for a certain region: Asian crisis in 1997 (for emerging markets), USD weakeningo 9/11 terror attack, Enron reporting scandal, Yukos case, etc.o Advantage: keep correlation between different factors

Hypothetical scenarioso Rising correlations: e.g., model each asset’s return Ri as wtd sum of Ri & market return RM

o Russia: bank crisis, inflation, WTO entry, …o It is important to ensure that there no internal inconsistencies in the perturbed model (e.g.,

arbitrage opportunities)o Advantage: very flexible

Hybrid methodo Change the covariance matrix & other parameterso Estimate the conditional distribution of ptf value and compute ‘stress’ VaR

Extreme-value theoryo Estimate the parameters of distribution of maximum losseso Block maxima: e.g., annual highest losseso Peak-over-threshold: 1% days with the lowest returno But: needs large and representative data base

Examples of scenarios Stock market falling (e.g., US in 1987)

o Developed stock markets falling by 20%, emerging ones by 30%; volatility rising by 25%

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o Flight to quality leads to strengthening of the dollar (by 10% relative to EM)o Interest rates in DM falling, EM % go up by 0.5% (1%) for long (short) maturitieso Commodity prices falling due to fear of recession: oil price goes down by 5%

Tightening of the Fed’s monetary policy (to tighten up inflation, as in 1984)o Overnight rates rising by 1%, longer-maturity rates by 0.5%o Interest rates in other countries rising lesso Relative strengthening of the dollar (investors put more in dollar-denominated instruments)o Credit spreads risingo Stock markets falling by 3-5%, volatility rising

Stress testing practice Morgan Stanley: blue book JP Morgan Chase: VID (vulnerability identification)

o Re-estimation at least once a month, discussed by the top managerso Several macroeconomic crisis scenarioso Complementary scenarios for different market segmentso Conservative assumption: positions remain the same during the crisis

To be ready to the abrupt fall in liquidity Objective: be prepared to any possible financial crisis

o Even though we don’t know its probability of occurring

RiskMetrics: JP Morgan & Reuters (since 1994) Market risk measurement methodologies

o Delta-normal / historical / Monte Carloo Stress testing: VID (vulnerability identification)

Data on volatilities and correlationso Cash flow mapping to stock indices, currencies, and FI baskets

Software

Example: Orange County, $1.7 bln losses in 1994 due to rising % Leveraged purchases of interest rate derivatives financed with reverse repos

o Betting on the slope of the yield curve Jorion: at the end of 1994, one-year 5% VaR was about $1 bln

o Are the downside risks worth the upside?o Bad incentives for managers: either modest, above-market return or financial disaster

Miller and Ross: Orange County was not insolvent, with net assets of $6 bln, including $600 mln in cash (for $20 bln in total assets)o The bankruptcy could have easily been prevented!

Quiz Total vs selective risk management

o Value vs cash flow at risk VaR vs expected return and upside potential How to manipulate VaR? What is the relation between Basel and RiskMetrics VaR? Is it bad if the actual losses exceed VaR?

Beyond VaR

VaR assumptions

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Portfolio sensitivityo Risk factor coverageo (Non-)linearity

Distributionso Stable and exploitable relationshipso Fat tails and skewness

Arbitrary parameters

VaR drawbacks Implicit assumptions

o No intraday tradingo Risks are described by exposures to several risk factorso Usually, second cross-derivatives are neglected

The actual losses are unknown The measurement error may be large, esp. for a high confidence level Model risk Manipulation: incentive to choose

o Strategies neutral to given risk factorso Portfolios with very unlikely extreme losses

Not subadditive

Properties of a coherent risk measure ρ Monotonicity: ρ(X1) ≥ ρ(X2) for X1 ≤ X2

o Higher return implies lower risk Translation invariance: ρ(X+const) = ρ(X)-const

o Adding cash lowers risk by the same amount Homogeneity: ρ(λX) = λρ(X) for λ ≥ 0

o Increasing the ptf’s size leads to proportional increase in risk Subadditivity: ρ(X1+X2) ≤ ρ(X1) + ρ(X2)

o Ptf risk does not exceed the sum of its components’ risks

Alternative market risk measures Full distribution Higher moments

o Variance, kurtosis, etc. Partial moments

o Semi-variance: risks in the falling market Downside CAPM: beta measured on the basis of low returns only

o Expected shortfall (coherent!) Cash flow risk

Lecture 8. Liquidity risk

Market liquidity: the ability to open or close large positions without a strong effect on price

Dimensions of market liquidity Tightness: price deviation

o Observed / Realized / Effective spread Driven by inventory and adverse selection costs

Depth: potential supply and demando Trading volume

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Which market is more liquid, with short or long-term instruments?

How does liquidity change with the degree of “specialness”?


o Turnover rate (to volatility)o % non-traded dayso Amihud (2002): daily ratio of absolute return to trading volume

Daily price response associated with $1 of trading Resiliency: time necessary for price to recover Immediacy: the execution time

Market microstructure estimates of illiquidityThe Glosten-Harris model: Δpt = λqt + ψ[Dt-Dt-1] + εt Trade-by-trade data D: order sign

o Buyer (seller) initiated if the price is above (below) mid-quote: +1 (-1) q: signed trade quantity

o Positive for buyer-initiated trades ψ: fixed cost component λ: inverse market depth parameter

The Hasbrouck-Foster-Viswanathan model: Δpt = α + λqUt + ψ[Dt-Dt-1] + εt

qU: the unexpected signed trading volumeo Based on the model for q with five lags of q and five lags of Δpo Measures the informativeness of trades

Cost components, in % of the priceo Proportional: λ * avg trade size / avg closing priceo Fixed: ψ / avg closing price

Liquidity and asset pricing Lower transaction costs lead to lower equity premium for risk Brennan (JFE, 1996):

o Portfolios with higher estimated costs have higher expected return Unexplained by Fama-French model

o Higher spread implies lower returns! Spread is a bad measure of illiquidity

Amihud (2002):o Expected illiquidity implies higher expected returno Unexpected decrease in liquidity decreases the current prices

Determinants of the liquidity Asset characteristics

o Substitution (leading to concentration) vs complementarity effects Market microstructure

o Transaction costso Info transparencyo Trading system

Auction vs dealership markets: price-volume trade-off Behavioral factor

o Dominating market participantso Heterogeneity

Buy/sell, risk attitude, investment horizon

Specifics of the Russian stock market Liquidity is concentrated in blue chips

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Liquidity moved from RTS to MICEX Flight to liquidity (or quality) during the crises

o E.g., August 1998, August 2003, April 2004

Modeling market liquidity risk The actual price may differ from the current market price The effective spread depends on the transaction size and timing

o Harder to measure for the OTC marketo Increases during the crisis

Monte Carlo analysis: VaR adjustment, accounting foro Direction and size of positions

No more positive homogeneity of degree 1 Ideally, should know elasticity of price wrt volume

o Correlation between market dynamics and liquidity Asymmetry between bullish and bearish markets

Stress testing, accounting foro Margin requirements

Esp. if you hold a large portfolio with one brokero Risk limits

Low limits will soon require further sales to stop losses o The likelihood of systemic crisiso Typical scenario: a dealer stops to provide quotes

Funding liquidity (insolvency) risk Inability to fulfill the obligations because of the shortage of liquid funds Determinants:

o Current cash reserveso Ability to borrow or generate CFs

Sources:o Systemic

E.g., Russian gvt default in August 17, 1998o Individual

Change of a company’s (implicit) credit ratingo Technical

Unbalanced forward payment structure Uncertainty about future CFs

Example: LTCM Investment strategy

o Betting on convergence of spreads E.g., long position on the off-the-run Treasury bonds, short position in the on-the-run

(recently issued and more liquid) Treasuries In general, long (short) position in riskier (less risky) instruments

o High leverage, no diversificationo Brought net returns over 40% in 1995 and 1996o But: sensitive to market-wide liquidity

At the end of 1997:o After returning $2.7 bln to their investorso Balance sheet assets of $125 blno Off balance sheet notional amounts round $1 trln

Mostly nettable (swaps)

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o Positions with nominal value of $1.25 trln Addressing funding liquidity risk:

o Own capital of $4.8 bln 30 times leverage for BS sheet assets only!

o Credit line of $900 mlno Investors commit capital for at least 2 years

The crisis and its resolutiono August 1998: Russian default triggered “flight to safety”, all risk premiums rose

LTCM lost $550 mln in August 21 onlyo By the end of September, capital declined to $400 mln

LTCM was forced to liquidate positions to meet margin callso September 1998: 90% of the LTCM ptf was purchased by a consortium of 14 banks for $3.625

bln That prevented the danger that the default of LTCM would trigger many cross-defaults

o July 1999: redemption of the fund The issues raised

o Risk management at LTCM Role of stress testing

o Risk management at LTCM counterparties Interaction with a highly leveraged institution

o Supervisiono Moral hazard

Lectures 9-13. Credit risk

Credit risk: losses due to the counterparty’s failure to honour his obligations

Credit event Default: an obligation is not honored Payment default: an obligor does not make a payment when it is due

o Repudiation: refusal to accept claim as valido Moratorium: declaration to stop all payments for some period of time

Usually, by sovereignso Credit default: payment default on borrowed money (loans and bonds)

Insolvency: inability to pay (even temporary) Bankruptcy: the start of a formal legal procedure to ensure fair treatment of all creditors

Specifics of CR: Asymmetry: possibility of big losses

o “the most you can lose is everything” Rare occurrence Longer horizon Non-tradability of most loans

o Hard to measure correlations Limits at the transaction level Interaction with market risk Esp important for banks Larger reserves

o Basel: 4*VaR

21

How do securitization and credit derivatives influence modeling CR?


Examples Savings&loans crisis in the US in 1980s

o The restructuring cost the gvt $30 bln Defaults on Latin American countries’ debt in 1980s

o Restructuring in Brady bonds Defaults on corporate bonds

o Esp. junk bonds at the end of 1990s Accumulation of bad quality loans in Japanese banks Russia

o No default on corporate bonds so faro Yukos on the brink of bankruptcy

Measuring credit risk Basic components:

o Probability of default (PD)o Recovery rate (RR) / loss given default (LGD), RR + LGD = 1o Credit exposure (CE) / Exposure at default (EAD), CE = EAD

Credit loss: CLi = Di*LGDi*EADi

o Di: default indicator The credit loss distribution: CL = Σi[Di*LGDi*EADi]

o Highly skewed: limited upside, high downsideo Correlation risk: assuming independence will

underestimate riskso Concentration risk: sensitivity to the largest loans

Credit VaR = WCLα – E[CL]o Difference between expected losses and certain

quantile of losseso Expected losses covered by ptf’s earningso Unexpected losses covered by capital reserves

Modeling credit risk Internal approach: usually used by banks, focus on PD

o Classical solvency analysiso Market environmento Quality of the company’s managemento Credit historyo Credit product characteristics

External approach: using market data on stocks / bondsMV(CR)= f(loss distribution, risk premium)

o Need to measure both PD and RRo Requires liquid secondary market

Recovery rate Highly variable, neglected by research for a long time

o Fragmented and unreliable data Market value recovery:

o MV per unit of legal claim amount, short time (1/3m) after the default Settlement value recovery:

o Value of the default settlement per unit of legal claim amount, discounted back to the default date and after subtracting legal and administrative costs

22

What is CE for futures or short option?


Legal environment factorso Collateral or guaranteeso Priority class: collateralized, senior, junior, etc.o The bankruptcy legislation

Large cross-country differences: e.g., US and France more obligor-friendly than UK US bankruptcy procedures: ch. 11 (aim to restructure the obligor) vs ch. 7 (aim to liquidate

the obligor and pay off debt) Other empirically observed factors

o Industryo The obligor’s rating prior to defaulto Business cycle & average rating in the industry

Altman: US, 1971-1995o On average, about 40% with SD of 20-30%

Modelling RR: beta distribution with density f(x) = c xa (1-x)b o For mean μ and variance σ2: a=μ2(1-μ)/σ2, b=μ(1-μ)2/σ2

Credit exposure: the amount we would lose in case of default with zero recovery Only the positive economic value counts: current CEt=max(Vt, 0)

o Vt: credit portfolio’s value Current vs potential

o Expected (ECE) vs Worst (WSE), for a given confidence level Loans and bonds: direct, fixed exposures

o Usually, at par Commitments (e.g., line of credit): large potential exposure

o Usually, fraction of par OTC derivatives: variable exposures

o Usually, at current value with adjustment for market risk E.g., 90% quantile of the distribution

o Interest rate swap: Diffusion effect: increasing risk over time Amortization effect: decreasing duration

Traditional approaches to CR measurement

Credit ratings Independent agencies: S&P, Moody’s, Fitch Integral estimate of the company’s solvency based on PD (and RR)

o S&P: “general creditworthiness… based on relevant risk factors”o Moody’s: “future ability… of an issuer to make timely payments of principal and interest on a

specific fixed-income security” The rating procedure

o Comprehensive analysis, from micro- to macro-level Quantitative: based on financial reports Qualitative: evaluation of the management (business perspectives, risk attitude, etc.) Legal

o The rating committee decides by votingo The rating is reconsidered at least once a year

Long-term company ratingo Investment rating: from BBB (S&P), Baa (M’s)

Meant for conservative investors

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o Speculative ratingo Smaller gradations: 1/2/3, +/-, Outlook, Watch

“Through-the-cycle” approach: CR at the worst point in cycle over contract’s maturityo Does not depend on the current market environmento In contrast to the “point-in-time” approach: CR depending on the current macro environment

Short-term company rating Rating of specific instruments

Applications: Credit policy and limits Pricing the credit Monitoring and credit control Securitization

Survival analysis: actuarial approach to estimating PD Sample period from 20 years

o Need to accumulate default statistics (esp. for first-class issuers)o Need to account for different stages of the business cycles

For each rating: average PDo Equal weights: PD = # defaults / # companies (with a given rating)

Another approach: weights proportional to the issue’s volume Type of the bonds

o Bonds traded in the US market vs bonds of US companieso Straight bonds vs convertible/redeemable bondso Newly issued bonds vs seasoned bonds

Horizono Lower # observations for longer horizon

Measures of PD Marginal mortality rate (MMR) in year t

o Estimated PD in year t after the issuance Survival rate (SR)

o In year 1: SR1 = 1-MMR1

o During T years: SRT = (1-MMR1)*…* (1-MMRT) MR in year t given survival during t-1 years: MRt = SRt-1*MMRt

Cumulative MR during T years: CMRT = Σt=1:TMRt = 1-SRT

Average MR during T years: AMRT = 1-(1-CMRT)1/T

Estimation results Cumulative PD goes down with rating The highest MMR is for 4-5 years since the bond’s issuance

Critique of credit ratings Ratings react with a lag to the change in the company’s solvency Bias in ratings: too conservative Large measurement error

o Companies with the same (different) rating may have different (same) ratingo Ratings of different agencies often do not coincide

Monte Carlo analysis by KMVo Generate 50,000 times a 25 year sample of N companies with a given PD

The parameters match those in Moody’s data base

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o The estimated PD is very noisy, higher than the actual one for most issuerso The differences between the estimated and actual PD rise with correlation between defaults

Credit ratings vs credit spreads Credit spread of a given bond often differs from the avg in the group with the same credit rating Credit spread depends on CR and other factors:

o Liquidity risko Maturityo Macroeconomic factorso Market volatility

Ratings react with a lag to the change in credit spreado Though the difference usually disappears with time: change in the firm’s solvency is reflected

in rating within half a year, or spread returns to the initial value

Internal rating systems: evaluation of the borrower’s solvency Criterions

o Probability of defaulto Recovery rate

Horizono Usually, 1 year

Scaleo According to S&P/Moody’s or its own

Factorso E.g., 5C: Character, Capital, Capacity, Collateral, Cycle

Analysis of financial reports Current / quick liquidity, leverage, profitability, turnover ratios Backward-looking: using historical data

o Extrapolation of the past into the future gives imprecise forecast Esp. under high uncertainy

In Russia: little trust to fin reports o RAS is clearly inferior to IASo Often reports are corrupted

To misguide tax authorities, minority shareholders, banks, etc.

Role of expert’s opinion Visit to the company Personal contact with opt managers Human factor

Credit scoring models: predicting the default based on the borrower’s data Altman’s Z-score (1968): linear function of 5 variables

o X1: Working Capital to Assetso X2: Retained Earnings to Assetso X3: EBIT to Assetso X4: Market Value of Equity to Book Value of Liabilitieso X5: Sales to Assets

The sample: 66 companies, half of which defaultedZ = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4+ 0.999X5

Interpretation:o Z > 3: default is unlikely

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Is type 1 or type 2 classification error more important?


o 2.7 < Z ≤ 3: closer to the “dangerous zone”o 1.8 < Z ≤ 2.7: likely defaulto Z ≤ 1.8: very high PD

How to evaluate the model?o Type-1 error: default by the borrower who received a loano Type-2 error: predicting default for the good borrowero Results: more than 90% firms were correctly classified

Constructing a scoring model Financial variables:

o Profitability, income volatilityo Leverage and interest coverageo Liquidityo Capitalizationo Management quality

Methodology o The binary choice logit / probit modelo Discriminant analysis

Examples:o ZETA-model (1977): for big companies ($100 mln in assets)

7 variables instead of 5o EMS (emerging markets score): for emerging countries

Using the model calibrated by the US firms Adjusted for the risk of currency’s devaluation, industry specifics, competitive advantage,

presence of a guarantee, etc.

Critiqiue: pros and cons Simplicity Solves the problem of subjectivity of experts’ grades Usually assumes simple linear dependence Limited theory to explain the degree of each variable’s impact

o Danger of overfitting in-sample Need a good data-base

o Usually, a biased sample excluding borrowers that were denied credit Same old problems with financial report data

How to estimate losses of a credit portfolio?

Requirements to an ideal CR model Instrument characteristics

o Seniorityo Collateral / guarantee

Company characteristicso Financial leverageo Balance liquidityo Type of the business: cyclical, growingo Size

Impact of macro factorso Recessiono Decline in the stock or commodity market

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o Market liquidity crisis Portfolio effects

o Interaction between different credit instrumentso Sensitivity to common risk factors

Basel requirements The internal CR models should capture

o Concentration / spread / downgrade / default risk Capital charge = 4*VaR(99%, 10d) Interaction between MR and CR

o Spread risk includes both interest rate risk (MR) and default premium (CR)o Credit events are often anticipated and reflected in spreado Default is a special case of downgrade

Ideally: integrated approach to MR and CR measurement

Classification of CR models Aggregation

o Bottom up: start from individual positionso Top down: model aggregate credit portfolio

Type of CRo Default-mode (loss-based): an exposure is held till maturity, when it is repaid or defaultso Mark-to-market (NPV-based): track changes in the current market value of the instrument

Method of estimating PD: o Conditional: depending on the current stage of the business cycleo Unconditional

Modeling defaulto Structural: (endogenous) decision by the firmo Reduced-form: (exogenous) stochastic process

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Proprietary models of credit risk

Benchmark CR models CreditMetrics, 1997

o JP Morgan Chase KMV Portfolio Manager, 1998

o Kealhofer, McQuown, Vasicek CreditRisk+, 1997

o Credit Suisse Credit Portfolio View, 1997

o McKinsey

CreditMetrics / Credit VaR I: CR driven by change in credit rating CR is modeled as change in credit rating during a period equal to VaR horizon

o Each rating is characterized by a certain PD Assuming that all companies with same rating have same risk of default

o Use migration probabilities (of moving from one to another rating)o Use data from one of the ratings agencies or internal ratings

Compute a probability distribution of the future value of the instrument, usingo Forward rates for each rating

Thus, account for duration and spread effectso Recovery rate in case of default

Depending on seniority Estimate VaR as a difference between a given quantile and expected value Bottom up approach:

o First estimate VaR for each instrumento Then aggregate to ptf VaR accounting for correlations

Migration of credit ratings Transition matrix, usually for 1 year

o Probability that a company with rating X next year receives rating Y How to estimate T-year transition matrix?

o Directly But: increasing T means fewer observations and larger measurement error

o Cross-multiplying annual transition matrices T times But: ignoring auto-correlation effects

Example: Bond with BBB rating, senior, unsecured, maturing in 5 years, 6% coupon rate

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Ratings: S&Po 7 groups: from AAA (first-class borrowers) to CCC (default)

Horizon: 1 yearo Could be from 1 to 10 years

Annual forward curve for each ratingo Allows us to compute the value of any bond in 1 year from now

Price of the bond in 1 year, if it keeps BBB rating

Forward distribution of the bond’s price

0.01 quantile of ΔV distribution: -23.91=VaR99%

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CreditMetrics for a portfolio: diversification effect Assume that return on assets is distributed as stock return (GBM) Determine thresholds for the return distribution corresponding to the actual migration probabilities Derive the correlation matrix

o Based on multifactor model with user-defined country and industry weights Monte Carlo analysis

o Generate joint rating migration scenarios for bonds within the portfolioo Estimate the empirical distribution of ptf value and compute VaR

Critique Ignore MR

o Derivatives require stochastic interest rateso Esp. important to assume stochastic interest rates for derivatives

Assume homogeneity with the same rating class Discrete migration matrix based on avg historical frequencies

o Transition probabilities are usually underestimated Simplified estimation procedure for correlations

Theoretical approaches to CR measurement

Structural approach: estimate risk-neutral PDo Use risky debt priceso Use equity prices and Merton (1974) option model

Reduced-form approach: use actuarial estimates of PDo Apply on the portfolio level

Credit spread analysis (based on bond’s market price) For a one-period zero-coupon bond: r-rf ≈PD*LGD

o LGD: relative to face value Components:

o Credit risk premium Usually rising with maturity

o Liquidity risk premium Determinants:

o Macro factors Market volatility / liquidity

o Bond characteristics

Merton model: stock as a call option on the value of the company V

Assume that V follows a log-normal distribution:o Zt~N(0,1): std Brownian motiono σV: volatility (SD) of the relative growth in V (dV/Vt)o μ: average growth rate, E[Vt]=V0eμt

The company’s capital structure includeso Equity, with value Eo Debt (zero-coupon), with face value F and maturity T

Default occurs at maturity if VT<Fo Stockholders receive at T: max(VT-F,0)

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o Creditors receive at T: min(VT, F) Stockholders: call option on the value of the company V

o Exercise date: To Price of the underlying asset: V o Volatility: σV

Creditors: bond and short puto Or basic asset (the company) and short call

Derivation of the parameters: V and σV are unobservable, derived from two equations:

o The value of equity by Black-Scholes: E=V*N(d1)-Fe-rTN(d2) r: risk-free rate d1=[ln(V/F) + T(r+σ2/2)] / [σV√T], d2=d1-σV√T

o The equation for stock volatility: σEE = N(d1) σVV Risk-neutral probability of default: PD = 1-N(d2) = N(-d2) Recovery rate (as % of the assets): RR = [1-N(d1)] / [1-N(d2)]

o The current market value of debt: V-E=e-rT[(1-PD)+PD*RR]*F

Implicit assumptions: Stockholders’ behavior – as given

o Though they are interested in raising risk Lognormal distribution

o Underestimate PD at short horizon Bankruptcy when V below the face value of debt

o Default may be different from bankruptcy

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KMV: main idea Estimate PD using the modified Merton model Move from empirical PD to risk-neutral PD Derive analytically the future value of the company’s obligations and VaR

KMV Credit Monitor (1993): EDF model Estimate the market value and volatility of the firm’s assets using modified Merton model

o For public companies: estimate V and σV based on equity prices Assume more complicated capital structure: equity, short-term debt, long-term debt, and

convertible preferred shareso For private companies: estimate V and σV based on financial accounting measures

V is between the operational value (proportional to EBIT) and book value σV is an empirical function of sales, assets, industry, etc.

Compute distance to default (measure of default risk in T years)o Empirically estimated default point: DP = short-term liabilities + ½ long-term liabilitieso Distance to default: DD = ln[E(V)-DP]/σV (in %, in σ)

Mapping DD to actual PD using historical data, for a given time horizono Expected Default Frequency: EDF = # defaulted companies / # companies, for given DDo Can compute implied rating

Critique Company-specific Continuous, not biased by periods of high or low defaults The correlation between defaults based on stock price correlations Testing: EDF rises sharply 1-2 years before the default

o Agencies’ ratings are slow to react Best applied to publicly traded firms

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o Can’t estimate country risk Ignore more complicated features of the debt

o Seniority, collateral, etc.

KMV Portfolio Manager: CR driven by change in MV(assets) Estimate actual EDF: KMV Credit Monitor Derive risk-neutral EDF

o Cumulative risk-neutral EDF at horizon T: QT = N[N-1(EDFT)+Sharpe√T)]o Substitute Sharpei=ρi*SharpeM*Tθ

In theory, θ=½ o Calibrate the market Sharpe ratio and θ with observed corporate spreads over LIBOR

For maturity t: r-rf=(-1/t)ln(1-Q*LGD) Estimate stock return correlation matrix using a 3-level multifactor model

o Individualo Country and industryo Global and regional

Portfolio’s losses: L=VT(NoDefault)-VT(equilibrium)o Analytical derivation of VaRo The limiting loss distribution: normal inverse (highly skewed and leptokurtic)

Critique Theoretically sound approach

o Using risk-neutral probabilities EDF is a good measure of default risk Need market prices of equity

o Assuming liquid market

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Why is actual st. deviation of PD larger than √μ?


o Can’t estimate country risk Hard to account for different types of debt Behaviour of equityholders – as given

o Though the have incentives and are able to increase risk Assuming lognormal distribution

CreditRisk+: actuarial approach to estimate PD Time horizon: usually, 1y Assume

o For each loan, PD is small and independent across periodso No assumption about the causes of the default

PD for a ptf: Poisson distribution, P(n defaults) = μnexp(-n)/n!o Avg # defaults: μ=ΣiPDi

o St. dev.: √μ Assume stochastical mean PD: μ is gamma-distributed

o Otherwise, volatility of PD is underestimated Exposure = Forward value * LGD

o Differing exposure amounts may result in a loss distribution far from Poisson The loan ptf is divided into exposure bands

o Each band j has same exposure νj (in rounded units)o For each band, expected loss: εj=νjμjs

Deriving analytical distribution of ptf losses:o Probability generating function for each bando Probability generating function for the entire ptfo Loss distribution of the entire ptf

Depends on two sets of parameters: εj and νj

Extensions:o Hold-to-maturity horizon

Decompose the exposure profile over timeo Multiple years

Take into account that default happens only onceo Sector analysis: dealing with concentration risk

Assign sector-specific parameters to given obligorso Scenario analysis

Critiqueo Easy implementation

Focus on default Few inputs

o Analytical form of the resultso Ignore MR and migration risko Reduced-formo Not applicable to non-linear instruments

Credit Portfolio View: top down approach using macro factors Assume PD(grade) = logistic f(macroeconomic index)

o The index = linear f(macro and industry factors)o Each factor follows AR(2)

%, FX, industry growth Estimate conditional rating migration matrix

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o Adjust the unconditional migration matrix by the ratio of conditional and unconditional simulated PD

o Recession: more mass to downgrade migrations and PD Monte Carlo analysis

o Simulate the joint cond distribution of default and migration probabilities Critique

o Link macro factors to default and migration probabilitieso Need reliable historical data on PD o Best applied to speculative grade obligorso Ad hoc procedure of estimating the migration matrix

Other models CreditGrades: RiskMetrics, Goldman Sachs, JP Morgan, Deutche Bank

o Stochastic default barrier: lognormally distributed, with possible discrete jumps Increases estimated PD

o PD is a closed form function of 6 parameters: Mean and std of RR Initial and current stock price Implied stock volatility: calibrated from actual CDS spreads

o CreditGrade = model implied 5y credit spread Algorithmics Mark-to-Future (MtF): scenario-based approach

o Links CR, MR, and LRo Generates cumulative PD conditional on scenario

Credit risk management

Impact of CR on derivatives Derivatives on (credit) risky bonds

o Long put gains in value Vulnerable derivatives: subject to CR by the writer

o The value diminishes by the credit spread: V’=V*Prisky(T)/PRF(T)

Traditional methods of CR management: credit exposure vs default modifiers Modeling CE

o BIS approach: CE = MV(deal) + potential CE Usually, potential CE as 0-15% of par

o Direct estimation of the potential CE distribution: Stress-case / trees / Monte Carlo

Collateral arrangements: become popular for (longer-dated) swapso Derivatives: usually, initial margin=0, variation margin only in case of MtM moves over the

exposure limito Longer period between remarkingso Post securities as collateral

Remarking / recouponing (to bring the value back to 0)o Usually quarterly

Netting: reduce potential CE to net positiono Better assessed by scenario and simulation analysiso But: risk of “cherry-picking”

Credit guaranteeso Esp valuable if the guarantee is from an independent company

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o Usually provided by parent companies Credit triggers

o Rating downgrade clause: right to terminate all transactions if the counterparty’s rating falls below the trigger level

Mutual termination optiono Time put: right to terminate on one or more dates using a pre-agreed formula to value the

transaction at these timeso Termination brings CE to 0

Modeling recovery rateo Derivatives: usually, rank equally with senior unsecured debt

Implementation issues Legal considerations

o Collateral: maybe problems with enforceability in case of bankruptcy Economic considerations

o Resources required to implement recouponing and collateral arrangements

CR management instruments Limits

o Maturity / collateral / currency / regional / industry Target levels

o Return to CR Diversification Reserves Credit derivatives

Credit derivatives

Evolution of derivatives: 3 waves Second and third generation of price and event derivatives

o Hybrid / contingent / path-dependent risks Strategic management

o Ptf risk, balance sheet growth, overall business performance New underlying risks

o Catastrophe / electricity / inflation / credit

Why credit derivatives? Separate CR mgt from the underlying asset

o Confidentialityo Customized terms o Objective and visible market pricing

Short-selling CR becomes possibleo Arbitrage and efficient markets

Off-balance-sheet operationso Banks can avoid selling loans

Tax considerations / underpricing / cliento Hedge funds can invest in CR, with leverage

Avoid transfer of property rights and administrative costs Completing the market

o Risk managers hedge CR

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What is a better hedge of CR: credit option on floating- (FRN) or on fixed-rate note?


o Issuers minimize liquidity costso Investors find interesting instruments

Rapid growth since end of 1990s, currently over $2 trln Types of insured CR:

o Credit event, value of the underlying asset, recovery rate, maturity Classified by:

o Type of the underlying asseto Trigger evento Payoff function

Credit default swap: Regular premium payments in exchange for a one-time premium in case of the credit event

o Materiality clause: credit event is not triggered by a technical defaulto Both credit event and payments can be linked to a group of obligations

Settlemento Fixed paymento Cash settlement: difference between the strike (par) and current market price o Physical settlement in return of the par amount

Basket default swapo The underlying asset: loan portfolio

First-to-default (basket) swap Dynamic credit swap

o Changing principal Practical role

o Enhance liquidityo Hedging / investment opportunity

Total return swap: Fixed or floating payments in exchange for the current income from the underlying asset

o Regular exchange of paymentso Insures both MR and CR

Credit options: Call or put on the price of FRN, bond, loan, or asset swap package

o (Multi-)European / Americano Can knock out upon credit event

Practical roleo Yield enhancement o Credit spread protectiono Hedging future borrowing costs

Downgrade options

Other credit derivatives Hybrid

o Require a material movement in %, equity prices, … Credit spread forwards / options Credit-linked note: coupon payments conditional on the credit event

o Usually via SPV (special purpose vehicle), trust company

Risks of credit derivatives

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Correlation: simultaneous default of the underlying asset and protection seller Basis Legal

o 1999, 2003: ISDA adopted standard terms and documentation Liquidity: usually traded OTC Protection:

o Bilateral nettingo Option for premature abortion of the contract in the case of the counterparty’s financial

distress

Lecture 14. Operational risk

Operational risk: Definition 1: financial risks besides MR and CR

o But: includes business riskso Did the credit default result from the ‘normal’ credit risk or mistake of the loan officer?

Definition 2: risks originating at financial transactionso But: excludes risks due to internal conflicts, model risk, …

Definition 3: risks due to deficiencies or mistakes fromo Information systems and technologieso Internal procedureso Personnel o External events

Classification of OR Operational failure (internal) risk

o People: incompetency / fraud o Process: model / transaction / operating control risko Technology: info systems, software, data bases

Operational strategic (external) risko Environmental factorso Change in political and regulatory regime

Usually, business risk is excludedo Choice of strategyo Loss of reputationo Legal

Examples of OR failures Barings and Nick Leeson (the “rogue trader”): more than $1 bln losses in 1995

o Strategy: cash-futures arbitrage, Singapore-Osaka arbitrageo Booked losing trades to account 88888o End of 1992: hidden losses of 2 mln poundso End of 1994: hidden losses of 208 mln poundso 1994: unauthorized positions in optionso January 1995: earthquake in Kobe, massive margin calls

Daiwa bank and Iguchi Toshihideo Since 1979, VP in NY, average profit of $4mln o 1984: loss of $50,000-200,000, o 1996: losses accumulated to $1.1 bln, confessedo The bank hid this from Fed, was fined $340 mln and had to close its US operations

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Sumitomo corporation and Hamanaka Yasuo, $2.6 bln total losseso Since 1975, in copper section, at heyday nicknamed “Mr. 5%”, “The Hammer”o From 1984: unauthorized speculative futures trading together with the head of the copper

trading team, trying to boost profitabilityo 1987: cumulative losses $58mln, Hamanaka became head of the copper section, received

$150 mln from Merrill Lyncho 1990: began borrowing money against Sumitomo’s trading stocks to fund his trading

positions, started fictitious option trades to create an impression of successo 1991: asked a broker to issue a backdated invoice for fictitious trades, worth $350 mln

The exchange notified Sumitomo, which replied it was needed for tax reasonso 1993: borrowed $100 mln from ING using forged signatures of senior managerso 1994: raised $150 mln from Morgan, then $350 mln from a 7-bank consortiumo 1995: investigations by US and UK regulators into unusual fluctuations in copper priceso March 1996: Sumitomo discovered that a statement from a foreign bank did not match its

recordso Hamanaka was jailed for 8 years, Sumitomo paid a fine of $150 mln in the US and $8 mln

in the UK, Merrill Lynch paid a fine of $15 mln in the US and $10 mln in the UK, o Sumitomo filed suits against Morgan and other banks in assisting the illegal trades

Specifics of OR Company-specific Hard to quantify Inverse relation between E(loss) and Prob(occurrence)

o HFLS: high frequency, low severity VaR techniques

o LFHS: low frequency, high severity Extreme value theory

Intentional (fraud) vs unintentional (mistakes) Interaction with MR and CR

Quantification of OR: top down vs bottom up approaches Indicators

o Key performance indicators E.g., # wrong operations

o Key control indicators: E.g., # prevented mistakes

o Key risk indicators Forecast OR based on performance and control indicators

Analysis of P&L volatility unexplained by MR and CR Causal models

o Measure losses using conditional probabilities Distribution of P&L

Management of OR Internal control system

o General policy by top managemento Assessment of riskso Control procedures

Internal: no conflicts of interests, double checking, approving access External: confirmation, audit

o Current monitoring

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Financial transactions systemo Front-office: “the face of the company”o Back-office: execution of the deals, accountingo Middle-office: input data, prepare papers, evaluate risks

Information system: o Data bases, softwareo Security, aggregation, interaction

New Basel agreement: reserves for OR Basic indicator approach (BIA): ORC = αGI

o GI: gross income, 3y avgo α: reservation coefficient, 15%

The standardized approach (TSA): ORC = Σi βi GIi o 8 std directions: corporate finance, trading operations, payments (all 18%), retail (12%) and

commercial banking (15%), intermediation (15%), asset management, brokerage (12%)o Alternative standardized approach: loans and advances instead of GI

Advanced measurement approaches (AMA) o Internal measurement approach (IMA): ORC = Σi γi ELi

Expected losses instead of GI based on PD, LGD, and correlations Can be adjusted for risk profile index (RPI)

o Loss distribution approach (LDA): ORC = Σi OVaRi

o Scorecard approacho Up to 20% of OR exposure can be insured

Special risks Model risk

o Wrong modelo Missing risk factoro Inputs

Low liquidity Legal risk

o Standardization 1992: ISDA Master agreement

Accounting risko Marking-to-market

Low liquidityo Hedging vs speculativeo Swapso Taxation

Integrated risk-management

Recent developments Increase in volatility after 1973 Globalization Deregulation Huge growth of the (exotic) derivatives market Higher volume of the off-balance (derivatives) operations Securitization Technological progress

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o Electronic trading systemso Program trading

Conclusions:o Aggregation of risks

Importance of the enterprise-wide RM Need unified framework

o Higher systemic riskso Higher operational risks

RAROC (Bankers Trust, end of 70s) Most popular risk-adjusted performance measure

o Other: RORAC, RARORAC RAROC = [Earnings – E(Loss)] / RC

o Earnings: profit net of all taxes and expenseso Expected losses:

CR: f(PD, CE, RR) MR: based on VaR models

o Risk capital: reserves covering losses with given prob for given horizon Can be adjusted with stress testing

Horizon: usually annualo Trade-off between MR and CR

Identification of riskso Usually: MR, CR, and ORo Additional: business, event, balance risks

Aggregation o Standard: RC = MRC + CRC + ORCo Monte Carlo

Applications of RAROC Enterprise level:

o Evaluate efficiency of work (backward-looking)o Optimal capital distribution (forward-looking)o Information for the outside world (shareholders, regulators, rating agencies)o Managerial compensation

Critiqueo Common bottom-up approacho Based on total risk (contrary to CAPM)o Inapplicable to risk-free instruments (unless impose positive reserves)

Regulation of banks The standardized framework vs internal models

o Internal models should satisfy certain criteria and be approved by CB Basel Capital Accord (BIS I, 1988)

o Differentiate the CR exposures The 1995 amendment

o Incorporate MR The new Basel Capital Accord (BIS II, 2003)

o Integrate MR, CR, and OR

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notes

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